Properties

Label 55.2.e.b.43.1
Level 55
Weight 2
Character 55.43
Analytic conductor 0.439
Analytic rank 0
Dimension 4
CM discriminant -11
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 55 = 5 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 55.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 43.1
Root \(-1.65831 - 0.500000i\)
Character \(\chi\) = 55.43
Dual form 55.2.e.b.32.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.15831 - 2.15831i) q^{3} -2.00000i q^{4} +(1.65831 + 1.50000i) q^{5} +6.31662i q^{9} +O(q^{10})\) \(q+(-2.15831 - 2.15831i) q^{3} -2.00000i q^{4} +(1.65831 + 1.50000i) q^{5} +6.31662i q^{9} +3.31662 q^{11} +(-4.31662 + 4.31662i) q^{12} +(-0.341688 - 6.81662i) q^{15} -4.00000 q^{16} +(3.00000 - 3.31662i) q^{20} +(2.84169 + 2.84169i) q^{23} +(0.500000 + 4.97494i) q^{25} +(7.15831 - 7.15831i) q^{27} -9.94987 q^{31} +(-7.15831 - 7.15831i) q^{33} +12.6332 q^{36} +(1.47494 - 1.47494i) q^{37} -6.63325i q^{44} +(-9.47494 + 10.4749i) q^{45} +(-9.31662 + 9.31662i) q^{47} +(8.63325 + 8.63325i) q^{48} -7.00000i q^{49} +(3.63325 + 3.63325i) q^{53} +(5.50000 + 4.97494i) q^{55} +3.31662i q^{59} +(-13.6332 + 0.683375i) q^{60} +8.00000i q^{64} +(11.4749 - 11.4749i) q^{67} -12.2665i q^{69} -3.00000 q^{71} +(9.65831 - 11.8166i) q^{75} +(-6.63325 - 6.00000i) q^{80} -11.9499 q^{81} -9.00000i q^{89} +(5.68338 - 5.68338i) q^{92} +(21.4749 + 21.4749i) q^{93} +(-3.52506 + 3.52506i) q^{97} +20.9499i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + O(q^{10}) \) \( 4q - 2q^{3} - 4q^{12} - 8q^{15} - 16q^{16} + 12q^{20} + 18q^{23} + 2q^{25} + 22q^{27} - 22q^{33} + 24q^{36} - 14q^{37} - 18q^{45} - 24q^{47} + 8q^{48} - 12q^{53} + 22q^{55} - 28q^{60} + 26q^{67} - 12q^{71} + 32q^{75} - 8q^{81} + 36q^{92} + 66q^{93} - 34q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −2.15831 2.15831i −1.24610 1.24610i −0.957427 0.288675i \(-0.906785\pi\)
−0.288675 0.957427i \(-0.593215\pi\)
\(4\) 2.00000i 1.00000i
\(5\) 1.65831 + 1.50000i 0.741620 + 0.670820i
\(6\) 0 0
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) 6.31662i 2.10554i
\(10\) 0 0
\(11\) 3.31662 1.00000
\(12\) −4.31662 + 4.31662i −1.24610 + 1.24610i
\(13\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 0 0
\(15\) −0.341688 6.81662i −0.0882234 1.76004i
\(16\) −4.00000 −1.00000
\(17\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 3.00000 3.31662i 0.670820 0.741620i
\(21\) 0 0
\(22\) 0 0
\(23\) 2.84169 + 2.84169i 0.592533 + 0.592533i 0.938315 0.345782i \(-0.112386\pi\)
−0.345782 + 0.938315i \(0.612386\pi\)
\(24\) 0 0
\(25\) 0.500000 + 4.97494i 0.100000 + 0.994987i
\(26\) 0 0
\(27\) 7.15831 7.15831i 1.37762 1.37762i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −9.94987 −1.78705 −0.893525 0.449013i \(-0.851776\pi\)
−0.893525 + 0.449013i \(0.851776\pi\)
\(32\) 0 0
\(33\) −7.15831 7.15831i −1.24610 1.24610i
\(34\) 0 0
\(35\) 0 0
\(36\) 12.6332 2.10554
\(37\) 1.47494 1.47494i 0.242478 0.242478i −0.575396 0.817875i \(-0.695152\pi\)
0.817875 + 0.575396i \(0.195152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 6.63325i 1.00000i
\(45\) −9.47494 + 10.4749i −1.41244 + 1.56151i
\(46\) 0 0
\(47\) −9.31662 + 9.31662i −1.35897 + 1.35897i −0.483779 + 0.875190i \(0.660736\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 8.63325 + 8.63325i 1.24610 + 1.24610i
\(49\) 7.00000i 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.63325 + 3.63325i 0.499065 + 0.499065i 0.911147 0.412082i \(-0.135198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 5.50000 + 4.97494i 0.741620 + 0.670820i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.31662i 0.431788i 0.976417 + 0.215894i \(0.0692665\pi\)
−0.976417 + 0.215894i \(0.930733\pi\)
\(60\) −13.6332 + 0.683375i −1.76004 + 0.0882234i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 11.4749 11.4749i 1.40189 1.40189i 0.607785 0.794101i \(-0.292058\pi\)
0.794101 0.607785i \(-0.207942\pi\)
\(68\) 0 0
\(69\) 12.2665i 1.47671i
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) 9.65831 11.8166i 1.11525 1.36447i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −6.63325 6.00000i −0.741620 0.670820i
\(81\) −11.9499 −1.32776
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000i 0.953998i −0.878904 0.476999i \(-0.841725\pi\)
0.878904 0.476999i \(-0.158275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 5.68338 5.68338i 0.592533 0.592533i
\(93\) 21.4749 + 21.4749i 2.22685 + 2.22685i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.52506 + 3.52506i −0.357916 + 0.357916i −0.863044 0.505128i \(-0.831445\pi\)
0.505128 + 0.863044i \(0.331445\pi\)
\(98\) 0 0
\(99\) 20.9499i 2.10554i
\(100\) 9.94987 1.00000i 0.994987 0.100000i
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −7.94987 7.94987i −0.783324 0.783324i 0.197066 0.980390i \(-0.436859\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) −14.3166 14.3166i −1.37762 1.37762i
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −6.36675 −0.604305
\(112\) 0 0
\(113\) −12.1583 12.1583i −1.14376 1.14376i −0.987757 0.156001i \(-0.950140\pi\)
−0.156001 0.987757i \(-0.549860\pi\)
\(114\) 0 0
\(115\) 0.449874 + 8.97494i 0.0419510 + 0.836917i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 19.8997i 1.78705i
\(125\) −6.63325 + 9.00000i −0.593296 + 0.804984i
\(126\) 0 0
\(127\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) −14.3166 + 14.3166i −1.24610 + 1.24610i
\(133\) 0 0
\(134\) 0 0
\(135\) 22.6082 1.13325i 1.94580 0.0975346i
\(136\) 0 0
\(137\) −10.1082 + 10.1082i −0.863601 + 0.863601i −0.991754 0.128154i \(-0.959095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 40.2164 3.38683
\(142\) 0 0
\(143\) 0 0
\(144\) 25.2665i 2.10554i
\(145\) 0 0
\(146\) 0 0
\(147\) −15.1082 + 15.1082i −1.24610 + 1.24610i
\(148\) −2.94987 2.94987i −0.242478 0.242478i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −16.5000 14.9248i −1.32531 1.19879i
\(156\) 0 0
\(157\) 16.4749 16.4749i 1.31484 1.31484i 0.397043 0.917800i \(-0.370036\pi\)
0.917800 0.397043i \(-0.129964\pi\)
\(158\) 0 0
\(159\) 15.6834i 1.24377i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −17.9499 17.9499i −1.40594 1.40594i −0.779334 0.626608i \(-0.784443\pi\)
−0.626608 0.779334i \(-0.715557\pi\)
\(164\) 0 0
\(165\) −1.13325 22.6082i −0.0882234 1.76004i
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 13.0000i 1.00000i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −13.2665 −1.00000
\(177\) 7.15831 7.15831i 0.538052 0.538052i
\(178\) 0 0
\(179\) 21.0000i 1.56961i 0.619740 + 0.784807i \(0.287238\pi\)
−0.619740 + 0.784807i \(0.712762\pi\)
\(180\) 20.9499 + 18.9499i 1.56151 + 1.41244i
\(181\) −9.94987 −0.739568 −0.369784 0.929118i \(-0.620568\pi\)
−0.369784 + 0.929118i \(0.620568\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.65831 0.233501i 0.342486 0.0171673i
\(186\) 0 0
\(187\) 0 0
\(188\) 18.6332 + 18.6332i 1.35897 + 1.35897i
\(189\) 0 0
\(190\) 0 0
\(191\) 23.2164 1.67988 0.839939 0.542681i \(-0.182591\pi\)
0.839939 + 0.542681i \(0.182591\pi\)
\(192\) 17.2665 17.2665i 1.24610 1.24610i
\(193\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 0 0
\(199\) 19.8997i 1.41066i 0.708881 + 0.705328i \(0.249200\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) −49.5330 −3.49379
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −17.9499 + 17.9499i −1.24760 + 1.24760i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 7.26650 7.26650i 0.499065 0.499065i
\(213\) 6.47494 + 6.47494i 0.443655 + 0.443655i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 9.94987 11.0000i 0.670820 0.741620i
\(221\) 0 0
\(222\) 0 0
\(223\) 14.4248 + 14.4248i 0.965957 + 0.965957i 0.999439 0.0334825i \(-0.0106598\pi\)
−0.0334825 + 0.999439i \(0.510660\pi\)
\(224\) 0 0
\(225\) −31.4248 + 3.15831i −2.09499 + 0.210554i
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) 29.8496i 1.97252i −0.165205 0.986259i \(-0.552828\pi\)
0.165205 0.986259i \(-0.447172\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(234\) 0 0
\(235\) −29.4248 + 1.47494i −1.91946 + 0.0962143i
\(236\) 6.63325 0.431788
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 1.36675 + 27.2665i 0.0882234 + 1.76004i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 4.31662 + 4.31662i 0.276912 + 0.276912i
\(244\) 0 0
\(245\) 10.5000 11.6082i 0.670820 0.741620i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 27.0000 1.70422 0.852112 0.523359i \(-0.175321\pi\)
0.852112 + 0.523359i \(0.175321\pi\)
\(252\) 0 0
\(253\) 9.42481 + 9.42481i 0.592533 + 0.592533i
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 22.2665 22.2665i 1.38895 1.38895i 0.561405 0.827541i \(-0.310261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(264\) 0 0
\(265\) 0.575188 + 11.4749i 0.0353335 + 0.704900i
\(266\) 0 0
\(267\) −19.4248 + 19.4248i −1.18878 + 1.18878i
\(268\) −22.9499 22.9499i −1.40189 1.40189i
\(269\) 13.2665i 0.808873i −0.914566 0.404436i \(-0.867468\pi\)
0.914566 0.404436i \(-0.132532\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.65831 + 16.5000i 0.100000 + 0.994987i
\(276\) −24.5330 −1.47671
\(277\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(278\) 0 0
\(279\) 62.8496i 3.76271i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(284\) 6.00000i 0.356034i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000i 1.00000i
\(290\) 0 0
\(291\) 15.2164 0.892000
\(292\) 0 0
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 0 0
\(295\) −4.97494 + 5.50000i −0.289652 + 0.320222i
\(296\) 0 0
\(297\) 23.7414 23.7414i 1.37762 1.37762i
\(298\) 0 0
\(299\) 0 0
\(300\) −23.6332 19.3166i −1.36447 1.11525i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 34.3166i 1.95220i
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 24.4248 + 24.4248i 1.38057 + 1.38057i 0.843600 + 0.536972i \(0.180432\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −25.1082 + 25.1082i −1.41022 + 1.41022i −0.651981 + 0.758236i \(0.726062\pi\)
−0.758236 + 0.651981i \(0.773938\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −12.0000 + 13.2665i −0.670820 + 0.741620i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 23.8997i 1.32776i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −9.94987 −0.546895 −0.273447 0.961887i \(-0.588164\pi\)
−0.273447 + 0.961887i \(0.588164\pi\)
\(332\) 0 0
\(333\) 9.31662 + 9.31662i 0.510548 + 0.510548i
\(334\) 0 0
\(335\) 36.2414 1.81662i 1.98008 0.0992528i
\(336\) 0 0
\(337\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(338\) 0 0
\(339\) 52.4829i 2.85048i
\(340\) 0 0
\(341\) −33.0000 −1.78705
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 18.3997 20.3417i 0.990609 1.09516i
\(346\) 0 0
\(347\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.7414 13.7414i −0.731383 0.731383i 0.239511 0.970894i \(-0.423013\pi\)
−0.970894 + 0.239511i \(0.923013\pi\)
\(354\) 0 0
\(355\) −4.97494 4.50000i −0.264042 0.238835i
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −23.7414 23.7414i −1.24610 1.24610i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −13.5251 + 13.5251i −0.706003 + 0.706003i −0.965692 0.259690i \(-0.916380\pi\)
0.259690 + 0.965692i \(0.416380\pi\)
\(368\) −11.3668 11.3668i −0.592533 0.592533i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 42.9499 42.9499i 2.22685 2.22685i
\(373\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(374\) 0 0
\(375\) 33.7414 5.10819i 1.74240 0.263786i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 29.8496i 1.53327i −0.642082 0.766636i \(-0.721929\pi\)
0.642082 0.766636i \(-0.278071\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 17.8417 + 17.8417i 0.911668 + 0.911668i 0.996403 0.0847358i \(-0.0270046\pi\)
−0.0847358 + 0.996403i \(0.527005\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 7.05013 + 7.05013i 0.357916 + 0.357916i
\(389\) 36.4829i 1.84976i 0.380265 + 0.924878i \(0.375833\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 41.8997 2.10554
\(397\) −20.8997 + 20.8997i −1.04893 + 1.04893i −0.0501886 + 0.998740i \(0.515982\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −2.00000 19.8997i −0.100000 0.994987i
\(401\) −26.5330 −1.32499 −0.662497 0.749064i \(-0.730503\pi\)
−0.662497 + 0.749064i \(0.730503\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −19.8166 17.9248i −0.984696 0.890691i
\(406\) 0 0
\(407\) 4.89181 4.89181i 0.242478 0.242478i
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 43.6332 2.15227
\(412\) −15.8997 + 15.8997i −0.783324 + 0.783324i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.0000i 1.17248i −0.810139 0.586238i \(-0.800608\pi\)
0.810139 0.586238i \(-0.199392\pi\)
\(420\) 0 0
\(421\) 39.7995 1.93971 0.969854 0.243685i \(-0.0783563\pi\)
0.969854 + 0.243685i \(0.0783563\pi\)
\(422\) 0 0
\(423\) −58.8496 58.8496i −2.86137 2.86137i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −28.6332 + 28.6332i −1.37762 + 1.37762i
\(433\) 29.4248 + 29.4248i 1.41407 + 1.41407i 0.717241 + 0.696826i \(0.245405\pi\)
0.696826 + 0.717241i \(0.254595\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 44.2164 2.10554
\(442\) 0 0
\(443\) −28.7414 28.7414i −1.36555 1.36555i −0.866677 0.498870i \(-0.833748\pi\)
−0.498870 0.866677i \(-0.666252\pi\)
\(444\) 12.7335i 0.604305i
\(445\) 13.5000 14.9248i 0.639961 0.707504i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 39.0000i 1.84052i −0.391303 0.920262i \(-0.627976\pi\)
0.391303 0.920262i \(-0.372024\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −24.3166 + 24.3166i −1.14376 + 1.14376i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 17.9499 0.899749i 0.836917 0.0419510i
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −0.575188 0.575188i −0.0267313 0.0267313i 0.693615 0.720346i \(-0.256017\pi\)
−0.720346 + 0.693615i \(0.756017\pi\)
\(464\) 0 0
\(465\) 3.39975 + 67.8246i 0.157660 + 3.14529i
\(466\) 0 0
\(467\) 23.0581 23.0581i 1.06700 1.06700i 0.0694117 0.997588i \(-0.477888\pi\)
0.997588 0.0694117i \(-0.0221122\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −71.1161 −3.27686
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −22.9499 + 22.9499i −1.05080 + 1.05080i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000i 1.00000i
\(485\) −11.1332 + 0.558061i −0.505535 + 0.0253403i
\(486\) 0 0
\(487\) 26.4749 26.4749i 1.19969 1.19969i 0.225436 0.974258i \(-0.427619\pi\)
0.974258 0.225436i \(-0.0723806\pi\)
\(488\) 0 0
\(489\) 77.4829i 3.50390i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −31.4248 + 34.7414i −1.41244 + 1.56151i
\(496\) 39.7995 1.78705
\(497\) 0 0
\(498\) 0 0
\(499\) 19.8997i 0.890835i 0.895323 + 0.445418i \(0.146945\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 18.0000 + 13.2665i 0.804984 + 0.593296i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 28.0581 28.0581i 1.24610 1.24610i
\(508\) 0 0
\(509\) 3.31662i 0.147007i 0.997295 + 0.0735034i \(0.0234180\pi\)
−0.997295 + 0.0735034i \(0.976582\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.25856 25.1082i −0.0554589 1.10640i
\(516\) 0 0
\(517\) −30.8997 + 30.8997i −1.35897 + 1.35897i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −43.1161 −1.88895 −0.944476 0.328581i \(-0.893430\pi\)
−0.944476 + 0.328581i \(0.893430\pi\)
\(522\) 0 0
\(523\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 28.6332 + 28.6332i 1.24610 + 1.24610i
\(529\) 6.84962i 0.297810i
\(530\) 0 0
\(531\) −20.9499 −0.909147
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 45.3246 45.3246i 1.95590 1.95590i
\(538\) 0 0
\(539\) 23.2164i 1.00000i
\(540\) −2.26650 45.2164i −0.0975346 1.94580i
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 21.4749 + 21.4749i 0.921578 + 0.921578i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 20.2164 + 20.2164i 0.863601 + 0.863601i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −10.5581 9.55013i −0.448165 0.405380i
\(556\) 0 0
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 80.4327i 3.38683i
\(565\) −1.92481 38.3997i −0.0809774 1.61549i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) −50.1082 50.1082i −2.09330 2.09330i
\(574\) 0 0
\(575\) −12.7164 + 15.5581i −0.530309 + 0.648816i
\(576\) −50.5330 −2.10554
\(577\) −18.5251 + 18.5251i −0.771208 + 0.771208i −0.978318 0.207109i \(-0.933594\pi\)
0.207109 + 0.978318i \(0.433594\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 12.0501 + 12.0501i 0.499065 + 0.499065i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20.6834 20.6834i 0.853694 0.853694i −0.136892 0.990586i \(-0.543711\pi\)
0.990586 + 0.136892i \(0.0437113\pi\)
\(588\) 30.2164 + 30.2164i 1.24610 + 1.24610i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −5.89975 + 5.89975i −0.242478 + 0.242478i
\(593\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 42.9499 42.9499i 1.75782 1.75782i
\(598\) 0 0
\(599\) 36.0000i 1.47092i 0.677568 + 0.735460i \(0.263034\pi\)
−0.677568 + 0.735460i \(0.736966\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 72.4829 + 72.4829i 2.95173 + 2.95173i
\(604\) 0 0
\(605\) 18.2414 + 16.5000i 0.741620 + 0.670820i
\(606\) 0 0
\(607\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7.73350 + 7.73350i −0.311339 + 0.311339i −0.845428 0.534089i \(-0.820655\pi\)
0.534089 + 0.845428i \(0.320655\pi\)
\(618\) 0 0
\(619\) 1.00000i 0.0401934i 0.999798 + 0.0200967i \(0.00639741\pi\)
−0.999798 + 0.0200967i \(0.993603\pi\)
\(620\) −29.8496 + 33.0000i −1.19879 + 1.32531i
\(621\) 40.6834 1.63257
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −24.5000 + 4.97494i −0.980000 + 0.198997i
\(626\) 0 0
\(627\) 0 0
\(628\) −32.9499 32.9499i −1.31484 1.31484i
\(629\) 0 0
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −31.3668 −1.24377
\(637\) 0 0
\(638\) 0 0
\(639\) 18.9499i 0.749645i
\(640\) 0 0
\(641\) 23.2164 0.916992 0.458496 0.888697i \(-0.348388\pi\)
0.458496 + 0.888697i \(0.348388\pi\)
\(642\) 0 0
\(643\) −5.57519 5.57519i −0.219864 0.219864i 0.588577 0.808441i \(-0.299688\pi\)
−0.808441 + 0.588577i \(0.799688\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.05806 8.05806i 0.316795 0.316795i −0.530740 0.847535i \(-0.678086\pi\)
0.847535 + 0.530740i \(0.178086\pi\)
\(648\) 0 0
\(649\) 11.0000i 0.431788i
\(650\) 0 0
\(651\) 0 0
\(652\) −35.8997 + 35.8997i −1.40594 + 1.40594i
\(653\) −27.1583 27.1583i −1.06279 1.06279i −0.997892 0.0648948i \(-0.979329\pi\)
−0.0648948 0.997892i \(-0.520671\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) −45.2164 + 2.26650i −1.76004 + 0.0882234i
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 62.2665i 2.40736i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(674\) 0 0
\(675\) 39.1913 + 32.0330i 1.50847 + 1.23295i
\(676\) 26.0000 1.00000
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 35.2164 + 35.2164i 1.34752 + 1.34752i 0.888350 + 0.459167i \(0.151852\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) −31.9248 + 1.60025i −1.21978 + 0.0611425i
\(686\) 0 0
\(687\) −64.4248 + 64.4248i −2.45796 + 2.45796i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 26.5330i 1.00000i
\(705\) 66.6913 + 60.3246i 2.51174 + 2.27195i
\(706\) 0 0
\(707\) 0 0
\(708\) −14.3166 14.3166i −0.538052 0.538052i
\(709\) 19.0000i 0.713560i −0.934188 0.356780i \(-0.883875\pi\)
0.934188 0.356780i \(-0.116125\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −28.2744 28.2744i −1.05889 1.05889i
\(714\) 0 0
\(715\) 0 0
\(716\) 42.0000 1.56961
\(717\) 0 0
\(718\) 0 0
\(719\) 51.0000i 1.90198i 0.309223 + 0.950990i \(0.399931\pi\)
−0.309223 + 0.950990i \(0.600069\pi\)
\(720\) 37.8997 41.8997i 1.41244 1.56151i
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 19.8997i 0.739568i
\(725\) 0 0
\(726\) 0 0
\(727\) 31.4749 31.4749i 1.16734 1.16734i 0.184510 0.982831i \(-0.440930\pi\)
0.982831 0.184510i \(-0.0590699\pi\)
\(728\) 0 0
\(729\) 17.2164i 0.637643i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(734\) 0 0
\(735\) −47.7164 + 2.39181i −1.76004 + 0.0882234i
\(736\) 0 0
\(737\) 38.0581 38.0581i 1.40189 1.40189i
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) −0.467002 9.31662i −0.0171673 0.342486i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) 37.2665 37.2665i 1.35897 1.35897i
\(753\) −58.2744 58.2744i −2.12364 2.12364i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.899749 + 0.899749i −0.0327019 + 0.0327019i −0.723269 0.690567i \(-0.757361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 0 0
\(759\) 40.6834i 1.47671i
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 46.4327i 1.67988i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −34.5330 34.5330i −1.24610 1.24610i
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) −96.1161 −3.46154
\(772\) 0 0
\(773\) 33.6332 + 33.6332i 1.20970 + 1.20970i 0.971123 + 0.238581i \(0.0766824\pi\)
0.238581 + 0.971123i \(0.423318\pi\)
\(774\) 0 0
\(775\) −4.97494 49.5000i −0.178705 1.77809i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −9.94987 −0.356034
\(782\) 0 0
\(783\) 0 0
\(784\) 28.0000i 1.00000i
\(785\) 52.0330 2.60819i 1.85714 0.0930902i
\(786\) 0 0
\(787\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 23.5251 26.0079i 0.834348 0.922406i
\(796\) 39.7995 1.41066
\(797\) −26.6913 + 26.6913i −0.945455 + 0.945455i −0.998587 0.0531327i \(-0.983079\pi\)
0.0531327 + 0.998587i \(0.483079\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 56.8496 2.00868
\(802\) 0 0
\(803\) 0 0
\(804\) 99.0660i 3.49379i
\(805\) 0 0
\(806\) 0 0
\(807\) −28.6332 + 28.6332i −1.00794 + 1.00794i
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0